Approximation of the first eigenpair of the p(x)-Laplacian using WEB-spline based mesh-free method

Abstract

Our objective in this article is to develop weighted extended B-spline (WEB-spline) approximation of the first eigenpair of p(x)-Laplacian problem. One of the reason for choosing WEB-spline based mesh-free method for solving p(x)-Laplacian problem is that, it has additional advantages compared to usual finite element method. For example, it does not require any mesh generation procedure and hence eliminates the difficult, time consuming preprocessing step in finite element method. Also, it combines the computational advantages of B-splines and geometric flexibility of standard mesh-based elements. To compute the first eigenpair of p(x)-Laplacian problem, we examine the problem numerically by applying an inverse power method with three different minimization techniques and we compare the results obtained. Finally, we study the asymptotic behaviour of the eigenvalue problem of p(x)-Laplacian and validate the results comparing with those given in the literature.